Lemma 2.2 ([1]} (see also [2]})) 
Let \(E_1, E_2, \ldots , E_k\)  be events in an arbitrary probability space. Suppose that each event \(E_i\)  is mutually independent of the set of all other events \(E_j\)  but at most \(d\) , and that \(\mathbb {P}[E_i] \le P\)  for all \(1 \le i \le k\) . If
\(e d P \le 1\) 
